Isildur and his sons journey north towards Rivendell

But soon [Isildur] departed, and... he bore away the Ring, to be an heirloom of his house, and marched north from Gondor by the way that Elendil had come..., for he purposed to take up his father's realm in Eriador....

The Silmarillion, Of the Rings of Power and the Third Age

When [Isildur] at last felt free to return to his own realm he was in haste, and he wished to go first to Imladris; for he had left his wife and youngest son there, and he had moreover an urgent need for the counsel of Elrond. He therefore determined to make his way north from Osgiliath up the Vales of Anduin to Cirith Forn en Andrath, the high-climbing pass of the North, that led down to Imladris....

It was a long journey, but the only other way... was far longer.... He had no fear, save for weather and weariness....

[The] second year of the Third Age was waning when Isildur set forth from Osgiliath early in Ivanneth, 1 expecting to reach Imladris in forty days, by mid-Narbeleth, 1 ere winter drew nigh in the North....

With Isildur went his three sons, Elendur, Aratan, and Ciryon, and his Guard of two hundred knights and soldiers, stern men of Arnor and war-hardened. Of their journey nothing is told until they had passed over the Dagorlad, and on northward into the wide and empty lands south of Greenwood the Great. On the twentieth day, as they came within far sight of the forest crowning the highlands before them..., the sky became overcast and a dark wind came up from the Sea of Rhûn laden with rain. The rain lasted for four days; so when they came to the entrance to the Vales, between Lórien and Amon Lanc, Isildur turned away from the Anduin, swollen with swift water, and went up the steep slopes on its eastern side to gain the ancient paths of the Silvan Elves that ran near the eaves of the Forest.

So it came to pass that late in the afternoon of the thirtieth day of their journey they were passing the north borders of the Gladden Fields, marching along a path that led to Thranduil’s realm, as it then was.... 2 [Above] the distant mountains clouds were gathering.... The Dúnedain were singing, for their day's march was near its end, and three parts of the long road to Imladris were behind them.

1Yavannië 5, according to the Númenórean "King's Reckoning," still kept with little change in the Shire Calendar. Yavannië (Ivanneth) thus corresponded to Halimath, our September; and Narbeleth to our October.

2[At] the end of the Second Age [Oropher] dwelt in the western glens of the Emyn Duir, and his numerous people lived and roamed in the woods and vales westward as far as Anduin, north of the ancient Dwarf-Road (Men-i-Naugrim).